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\title{\textbf{Innovation, Renewable Energy, and Macroeconomic Growth\thanks{%
We thank Jane Kliakhandler for research assistance. We also thank audiences
at the Baker Institute Energy Forum, the Centro Estudios Monetarios Y
Financieros (CEMFI), Madrid, the Banque de France, the Banco de Portugal,
the Rice University Economics Department, and the University of Munich for
comments and suggestions.}} }
\author{Peter Hartley\thanks{%
Department of Economics, Rice University} \and Kenneth B. Medlock III\thanks{%
James A. Baker III Institute for Public Policy, Rice University} \and Ted
Temzelides\thanks{%
Department of Economics, Rice University} \and Xinya Zhang\thanks{%
Department of Economics, Rice University}}
\maketitle

\spacing{1.5}

\newpage

\begin{center}
\bigskip

\textbf{Extended Abstract}
\end{center}

\noindent Is the energy sector under-investing in R\&D? How would subsidies to renewable energy and taxes on fossil fuels affect
renewable energy adoption and macroeconomic growth? The answers to these
questions are important for several reasons that go beyond the standard
arguments about under-investment in R\&D in other industries. Energy is an essential input to much economic activity and as such is
viewed as one of the sectors that directly connects to national security. In
addition, there is an extensive ongoing policy discussion in the United
States about innovations in the \textit{Green Economy} and their potential
to act as a new engine of economic growth. With a large number of resources
potentially redistributed towards the renewable sector, it is important to
have such policies evaluated. This, in turn, requires building models in
which there is a clear link between innovation in renewable energy,
government taxes and subsidies, and economic growth.

There are some theoretical arguments, as well as certain empirical
indications, that R\&D in the energy sector is low in relative terms. The
theoretical reasoning can be developed around the notion of
\textquotedblleft \textit{creative destruction.}\textquotedblright\
Innovation often results in old technologies becoming obsolete. In the energy sector, this is exacerbated by regulatory uncertainty. Large fixed costs and long time horizons also mean that firms in the industry have a lot at stake when choosing investment plans. Profit maximization therefore might lead energy
companies to be reluctant to invest substantial resources into R\&D in
new technologies. This reluctance to adopt revolutionary changes, as opposed
to investing in improving the existing structure, might indeed lead to a
market failure, resulting in a discrepancy between the profit maximizing and
the socially efficient level of R\&D. This, in turn, might imply the need
for government subsidies or related measures, such as taxing fossil fuels,
that could induce additional R\&D in renewable energy, thus, speeding up the
transition towards a renewable energy based economy.

In the early 80s, energy companies in the US were investing more in R\&D
than drug companies. However, a negative trend has been observed since. By
comparison, China, which passed a landmark renewable law in 2007, has
pledged to spend \$200 billion on renewable energy over the next 15 years.
In an attempt to reverse the trend in the US, President Obama proposed
spending \$150B over the next 10 years on renewable energy R\&D compared to
the current \$5B/year. Some studies have argued that such investments are
likely to provide high returns for the society and to pay off in terms of
long run growth in output and employment.

With over two trillion dollars in
annual sales, the energy industry is the largest on the planet. Thus,
economic policies that affect the energy sector have global consequences.
Yet, seldom are such policies studied and evaluated using the standard tool
of macroeconomics: quantitative-dynamic-general-equilibrium modeling. In
this paper, we build a model in which to study technological progress in
renewable energy and its consequences for macroeconomic growth. We compute
the equilibrium optimal path of investment in both the fossil fuel and
renewable energy sectors and calibrate the model using data on world energy consumption and production and cost data from the US.
Finally, we use the model to evaluate different policy scenarios regarding
imposing taxes on the use of fossil fuel and offering government subsidies
(financed by taxation) on the use and development of renewable energy.

Absent any government intervention, we find that the economy goes through
three distinct regimes related to its energy needs. Initially, production
uses fossil fuel only, and investment takes place in order to improve the
efficiency of supplying fossil fuel. In the medium to long run, as the price
of fossil fuel inevitably increases, investment and capital accumulation in
the economy suffer. The economy then makes a transition to the first of two
renewable energy regimes. Here, learning-by-doing and investment in the
renewable technology know-how reduce the cost of producing energy using the
backstop technology. Finally, in the very long run, a transition to the
second renewable energy scheme occurs. Here, a limit is reached after which
renewable energy is produced at the lowest possible cost.

Next, we examine how this transition is affected by imposing taxes on fossil
fuel energy or subsidies on R\&D in renewable energy. Taxing fossil fuels
accelerates the rate of adoption of the renewable energy technology.
However, a main finding of our analysis is that the elasticity of the
adoption rate appears to be small. In our model economy, a tax as high as $%
20\%$ accelerates the renewable technology adoption by about eleven years,
while a more realistic $2\%$ tax accelerates the transition by only five
years. The tax leads to a less intensive fossil fuel use, but also creates a
distortion in the economy. Subsidies on renewable energy investment also
accelerate the rate of adoption of the renewable energy technology. Indeed,
a renewable energy subsidy appears to be more effective than a tax on fossil
fuels, with a $2\%$ subsidy accelerating the introduction of the renewable
energy regime by nineteen years. However, the renewable energy subsidy also
leads to a more intensive use of fossil fuel reserves in the short run. This
somewhat paradoxical conclusion could be important for policy makers. It can
be explained as follows. Since the abandonment of fossil fuel is accelerated
as a result of the subsidy, the opportunity cost of using fossil fuel in the
short run declines. Thus, while the subsidy on renewables leads to a faster
transition away from fossil fuels, it also implies a more intensive short
run use of fossil fuel than what is socially optimal. This could imply an
increase in emissions associated with fossil fuel combustion in the short run.

\noindent \emph{\ }\newpage

\section{Introduction}

Many studies assume that the optimal size of R\&D in the energy sector is 5
to 10 times the current level. Is the energy sector under-investing in R\&D?
What would be the effects of subsidies to R\&D in renewable energy on
macroeconomic growth? The answers to these questions are important for
several reasons that go beyond the standard arguments about under-investment
in R\&D in other industries. Energy is an essential input to much economic activity and as such is
viewed as one of the sectors that directly connects to national security. In addition, and perhaps more
importantly from an economic perspective, there is an extensive ongoing
policy discussion in the United States about innovations in the \textit{%
Green Economy} and their potential to act as a new engine of economic
growth. With substantial resources devoted to production
and investment subsidies in the renewable sector, it is important to have such
policies evaluated. This, in turn, requires building models in which there
is a clear link between innovation in renewable energy, government
subsidies, and Gross Domestic Product (\textit{GDP}) growth. This paper
attempts to provide such a model.

There are indications that R\&D in the energy sector is low in relative
terms. In the early 80s, energy companies were investing more in R\&D than
drug companies. However, a negative trend has been observed since. By
comparison, China, which passed a landmark renewable law in 2007, has
pledged to spend \$200 billion on renewable energy development over the next
15 years. In an attempt to reverse the trend in the US, President Obama
proposed spending \$150 billion over the next 10 years on renewable energy
R\&D compared to the current \$5 billion/year.

Several studies have argued that investments in new energy sources are
likely to provide high returns for society as a whole. Examples include Kammen and
Nemet (2005), who have argued that the economic benefit from a 5- to -
10-fold increase in energy R\&D spending over the current levels would repay
the country in job creation and global economic leadership, building a
vibrant, environmentally sustainable engine of new economic growth. Apollo
Alliance suggests that a major investment in alternative energy technologies
could add more than 3.5 million new jobs to America's economy, stimulate
\$1.4 trillion in new GDP, and pay for itself within 10 years. Finally, a
2008 University of Massachusetts study finds that a \$100 billion investment
in green programs would create about two million jobs over two years.%
\footnote{%
See Pollin et.al. (2008): \textquotedblleft Green Recovery: A program to
create good Jobs and Start Building a Low-Carbon Economy.\textquotedblright}

Other researchers reach more sober conclusions about the potential
effects of subsidies to renewable energy on economic activity and job
creation. An Universidad Rey Juan Carlos study on the Spanish
experience finds that a \$36 billion total subsidy to renewable
energy between the years 2000 and 2008 created an estimated 50,000
related jobs (mainly in construction, maintenance, operation, and
administration). However, the study concludes that the implied average
subsidy of \euro 571,000 per job in renewable energy led to an
estimated 9 jobs lost in the economy for every 4 created. This might
be attributed to the fact that any kind of excess government spending
due to subsidies is likely to lead to a combination of higher future
energy prices, taxes, and debt, which will themselves tend to reduce
employment. Subsidies also could \textquotedblleft
absorb\textquotedblright\ capital away from other, perhaps more
productive parts of the economy. Since none of these factors appears
to be unique to the Spanish experience, it would be worth
investigating their relevance to the US economy.

With over two trillion dollars in annual sales, the energy industry is the
largest on the planet. Thus, economic policies that affect the energy sector
are likely to have global consequences. Yet, seldom are such policies
studied and evaluated using the standard tools of macroeconomics:
quantitative, dynamic general equilibrium modeling.\footnote{%
See Kydland and Prescott (1982).} We build a model in which to study
technological progress in renewable energy as a potential engine of
macroeconomic growth. We compute the equilibrium optimal path of investment
in both the fossil fuel and the renewable energy sectors. Finally, we
evaluate different policy scenarios regarding imposing taxes on the use of
fossil fuel and offering government subsidies (financed by taxation) on the
use and development of renewable energy.

Our basic model involves a growth model in continuous time. As in Hartley
and Medlock (2005), energy is needed in order to produce the economy's
single consumption good. Energy can come from two sources: fossil fuel and a
renewable source. The marginal resource cost of fossil fuel extraction
increases with the total quantity of resources mined to date. At the same
time, we assume that investments in mining technology or energy efficiency
can reduce the unit cost of supplying fossil fuel. Turning to the renewable
source, we explicitly model technological progress by assuming that, due to
learning-by-doing, experience with using renewable energy lowers the unit
cost of these energy sources.

We find that the economy goes through three distinct regimes. Initially,
production uses only fossil fuel, and investment takes place in order to
improve the efficiency of supplying fossil fuel. In the medium to long run,
the price of fossil fuel inevitably increases, and the economy makes a
transition to a renewable energy regime. Here, renewable energy is used and,
at the same time, learning-by-doing reduces the cost of using the backstop
technology. Finally, in the very long run, a limit is reached after which
renewable energy is produced at the lowest possible cost.

We calibrate the model using data from the \textit{Energy Information
Administration} (EIA), the \textit{Survey of Energy Resources}, and the%
\textit{\ GTAP 7 Data Base} produced by the \textit{Center for Global Trade
Analysis} in the Department of Agricultural Economics, Purdue University.
The last mentioned data source provides a consistent set of international
accounts that also take account of energy flows.

We then examine how the transition to renewable energy is affected by imposing taxes on fossil fuel energy or by
imposing subsidies to renewable energy. Taxing fossil fuels accelerates the
rate of adoption of the renewable energy technology. However, a main finding
of our analysis is that the elasticity of the adoption rate appears to be
small. A tax as high as $20\%$ accelerates the renewable technology adoption
by about eleven years, while a more realistic $2\%$ tax accelerates the
transition by only five years. The tax leads to a less intensive fossil fuel
use. However, the resulting distortion creates a wedge between the
equilibrium and the socially optimal level of investment. As a result, it
can be shown that welfare in the economy declines in the tax size.

In our model, subsidies on renewable energy investment also accelerate the rate
of adoption of the renewable energy technology. Indeed, a renewable energy
subsidy appears to be more effective than a tax on fossil fuels, with a $2\%$
subsidy accelerating the introduction of the renewable energy regime by
nineteen years. As a result of the renewable energy subsidy, the fossil fuel
reserves are used more intensively in the short run. This somewhat
paradoxical conclusion can be explained as follows. Since the adoption of
renewable fuel is accelerated as a result of the subsidy, the opportunity
cost of fossil fuel use declines in the short run. Thus, while the subsidy
on renewables leads to a faster transition towards renewable energy, it also
implies a more intensive use of fossil fuel than what is socially optimal in
the short run. While we do not model carbon dioxide or other emissions explicitly in our
analysis, it is worth mentioning that this could imply a short run increase in
greenhouse gas and other emissions associated with fossil fuel combustion.

The paper proceeds as follows. The next section presents some stylized
facts, documenting the trends in R\&D and in innovation in the energy
sector. This involves studying (public) R\&D dollars and patents, as well as
\textquotedblleft learning curves\textquotedblright\ derived from data from
the renewable industry. Section 3 develops our main model. One of our main
methodological contributions involves embedding R\&D for fossil fuel
production, fossil fuel depletion costs, and learning curves for renewable
energy into a calibrated macroeconomic model of growth. We solve the
corresponding optimal growth problem and discuss how the model is calibrated
in order to perform numerical simulations. Finally, Section 4 studies the
effects of different policy scenarios regarding the optimal rates of
renewable technology adoption and the consequences for GDP growth. A brief
conclusion follows.

\section{Measuring Technological Progress}

\subsection{Patents}

Although energy produced using renewable sources currently is more
expensive than that produced by fossil fuel, the gap appears to be
shrinking. Eventually, rising costs of fossil fuel and falling costs of renewable energy will lead to a transition to a
predominantly renewable energy regime. Studying the determinants of when this
parity in energy costs will occur, and especially the possible effects of policy on the transition, is one of the main goals of our paper.

In order to evaluate the effects of policy, it is important to
understand the rate of technological progress in renewable energy in
the presence and in the absence of government policy. To begin with, how can one
measure technological progress in the renewable energy sector? One approach
involves counting the number of new patents filed
in the industry.\footnote{%
See, for example, Popp (2002).} To measure progress, however, each patent needs to be weighted by the importance of the innovation it represents. 
Since each submitted patent contains citations to earlier related patents, one measure of the extent of innovation stimulated by a patent is the number of times it is subsequently cited. A simple procedure for measuring progress, therefore, is to count the number of patents that have been cited at least a minimum number of times. Another measure only counts patents  that have
been renewed upon expiration, since renewal is another signal of the patent's value.

Figures~\ref{fig:WindR&D} and \ref{fig:WindPatents} plot data from 1976 to 2009 on public R\&D and
patents for wind energy that have been renewed and cited at
least 3 times. Figures~\ref{fig:SolarR&D} and \ref{fig:SolarPatents} plot
the corresponding data for solar energy.\footnote{%
Public R\&D (in 2008 \$) from the DOE. Patent data from the US Patent
Office. Solar includes PV and thermal. Note that the number of cited patents
inevitably declines towards the end of the sample period, as the newest
patents have fewer opportunities to be cited.}

\begin{figure}[ht]
\centering \includegraphics[width=4.0712in]{windRD3T.pdf}
\caption{Wind R\&D expenditure}
\label{fig:WindR&D}
\end{figure}

\begin{figure}[ht]
\centering \includegraphics[width=4.0712in]{windPAT3T.pdf}
\caption{Wind patents}
\label{fig:WindPatents}
\end{figure}

\begin{figure}[ht]
\centering \includegraphics[width=4.0712in]{solarRD3T.pdf}
\caption{Solar R\&D expenditure}
\label{fig:SolarR&D}
\end{figure}

\begin{figure}[ht]
\centering \includegraphics[width=4.0712in]{solarPAT3T.pdf}
\caption{Solar patents}
\label{fig:SolarPatents}
\end{figure}

These graphs document non-increasing trends in both public R\&D expenditure
and innovation, as measured by patents, in both solar and wind energy. At
least two explanations can be given for these trends. Several authors have
pointed to decreased energy prices and low R\&D budgets following the end of
the oil crisis in the 70s. This interpretation can find a theoretical
justification in the literature on endogenous growth and, more specifically,
the problem of \textquotedblleft \textit{creative destruction.}%
\textquotedblright\ This argument goes as follows. New technologies often
result in old ones becoming obsolete. Given the large fixed costs and the
regulatory uncertainty that are prevalent in the energy sector, private profit maximization might lead energy companies to be
particularly reluctant to invest substantial resources in renewable energy
R\&D. This reluctance to risk abandoning existing technologies might indeed
lead to a market failure, resulting in a discrepancy between the profit
maximizing and the socially efficient level of R\&D. This, in turn, might
imply the need for government policies that could induce additional R\&D in
renewable energy.

There is, however, another possibility. Declining levels of innovation
(patents) might be the result of having reached a \textquotedblleft
technological frontier\textquotedblright\ that, given the existing state of
knowledge, makes additional innovation very hard.\footnote{%
Popp (2002) notes that while energy prices did not peak until 1981,
patenting activity in a variety of renewable energy sources reached its peak
in the late 1970's. That patenting activity drops before prices might be an
indication of diminishing returns to R\&D.} In that case, subsidies might
only have a marginal positive effect since learning-by-doing, and the
corresponding passage of time, might be necessary before the renewable
technologies become truly competitive. Our study does not attempt to
discriminate between these two hypotheses. Instead, we will study the
optimal levels of innovation in renewable energy and the resulting length of
the transition to a \textquotedblleft green economy.\textquotedblright\ We
will also study how the length of this horizon is affected by imposing
different tax/subsidy schemes. Before we introduce our model, we will
discuss one alternative tool for measuring technological progress that we
will use in our analysis later: the\textit{\ learning (experience) curve}.

\subsection{Learning Curves}

An alternative way to measure technological progress involves the use of 
\textit{learning curves} These curves describe how
marginal costs decline with cumulative production. Typically, this
relationship is characterized empirically by a \textit{\textquotedblleft power
law\textquotedblright\ }of the form

\begin{equation}
P_{t}=P_{0}X^{-\alpha }
\label{eq:Learning}
\end{equation}%
\textit{\ }where $P_{0}$ is the initial price (\textit{\$ cost of} first 
\textit{MW of sales), }$X$ is the cumulative production up to year $t$, and $%
2^{-\alpha }$ is the \textit{Progress Ratio} ($PR$). For each doubling of
the cumulative production (sales), the cost declines to $PR\%$ of its
previous value. Taking logarithms on both sides results in a straight line
if logarithmic axes are used; i.e., $\ln P_{t}=\ln P_{0}-\alpha \ln X$. As
an example, Figure~\ref{fig:SolarLC} provides a learning curve for the price
of photovoltaic modules between 1976 and 1992.

\begin{figure}[tbp]
\centering \includegraphics[width=4.0712in]{PVmodulesKP.pdf}
\caption{PV modules (International Energy Agency, 2000)}
\label{fig:SolarLC}
\end{figure}

The apparent decline in costs might be due to several reasons, including
process innovation, learning-by-doing, economies of scale, product
innovation/redesign, input price declines, etc. Learning curves aggregate
these factors. Such tools can also be used to guide policy. As of today, no
renewable energy source is directly competitive to fossil fuel for widespread
energy production. However, we would expect the costs associated with
producing fossil fuels to increase over time as the most easily-mined
resources are depleted. On the other hand, the costs of renewable energy
should decline as a result of research investments and also as the volume of
renewable energy increases. At some point in the future, therefore, parity
will be reached. Proponents of government subsidies argue that, by
subsidizing renewable energy research, or renewable energy use, we can
hasten the decline in costs of renewable energy and, as a result, speed up the path towards parity.
It is important to note, however, that such subsidies involve several direct
and indirect costs.\footnote{%
When it comes to policy, it is worth mentioning that there is no reason to
believe that subsidies to renewable energy use \textquotedblleft bend
down\textquotedblright\ experience curves. At best, they could accelerate
progress along the curve. By contrast, direct R\&D subsidies might reduce
costs more directly. Nevertheless, even if such subsidies succeed in making
new energy sources more competitive, they are not necessarily worthwhile.
For example, we must also consider other costs associated with the switch,
such as the need to replace existing capital tied to current fuel sources,
the opportunity costs of subsidies, and the fact that learning-by-doing
takes time, not just volume of production. We need a general equilibrium
economy-wide model in order to quantify such opportunity costs and the
effects of subsidies on other sectors, as well as the intertemporal
benefits. The goal of this paper is to provide such a model.} Later in our
analysis we will imbed a version of an learning curve into a general
equilibrium macroeconomic model. This will allow us to study the effects of
innovation on economic growth. The next section introduces the main
ingredients of the model.

\section{The Macro Model}

\subsection{Production Technology}

We model economic activity in continuous time, indexed by $t$. The state variables, the controls, and the technology variables thus are functions of $t$. We shall usually simplify notation, however, by omitting
time as an explicit argument.

There is a
single consumption good in the economy. We assume that the per capita output
of the good can be written as a linear function of a per capita stock of
capital $k$:\footnote{%
We assume that technological progress allows the \textquotedblleft
productive services" supplied by inputs to expand even if the physical
inputs stop growing. In particular, we implicitly assume that labor input
can be expanded through investment in human capital even if hours and number
of employees remain fixed. Hence, the marginal product of capital does not
decline as $k$ accumulates.} 
\begin{equation}
y=Ak  \label{eq:ProdFn}
\end{equation}%
We assume that capital depreciates at the rate $\delta $, while investment
in new capital is denoted by $i$: 
\begin{equation}
\dot{k}=i-\delta k  \label{eq:kdot}
\end{equation}%
Energy is also needed to produce output. At each instant, the ratio of
energy to capital inputs is fixed. Denote the per capita energy derived from
fossil fuel resources that is used to produce goods by $R\geq 0$. Per capita renewable energy supplied by the backstop technology $B\geq 0 $ is a perfect substitute for the energy produced from fossil fuel burning. Thus, we assume that at each moment of
time: 
\begin{equation}
R+B=y  \label{eq:NrgInput}
\end{equation}%

Letting $c$ denote per capita consumption, we assume that the lifetime
utility function is given by: 
\begin{equation}
U=\max \int_{0}^{\infty }e^{-\beta \tau }\frac{c(\tau )^{1-\gamma }}{%
1-\gamma }\,d\tau  \label{eq:Objective}
\end{equation}

The term $e^{-\beta \tau }$ acts as a discount factor, capturing the fact
that utility from future consumption is less valuable than today's
consumption.

\subsection{Fossil Fuel Supply}

Let $Q$ denote
the (exogenous) population and labor supply and assume that it grows at the
constant rate $\pi $. The total fossil fuel used will then be $QR$.
Following Heal (1976) and Solow and Wan (1976), we assume that the most
easily-mined, or the richest deposits or fields, tend to be exhausted first.
The marginal resource costs of extraction then increase with the total
quantity of resources mined to date, $S$, which is also the integral of $QR$%
: 
\begin{equation}
\dot{S}=QR  \label{eq:Sdot}
\end{equation}%
Heal introduced the idea of an increasing marginal cost of extraction to
show that the optimal price of an exhaustible resource begins above marginal
cost, and falls toward it over time.\footnote{%
This claim is rigorously proved in Oren and Powell (1985).}

We modify the resource depletion model to also allow for technical change in
mining exploration. The marginal cost of extraction, $g(S,N)$, depends not
only on $S$ but also the state of technical knowledge $N$. It is useful to define the energy supplies in efficiency units. Improvements in energy efficiency then also will lead to reduction in the
per-unit cost $g(S,N)$ of supplying an additional unit of $R$. Investment in
mining technology, or the efficiency with which fossil fuel is used to
provide useful energy services, leads to an accumulation of knowledge: 
\begin{equation}
\dot{N}=n  \label{eq:Ndot}
\end{equation}%
We then assume that $g(S,N)$ is given by the following function: 
\begin{equation}
g(S,N)=\alpha _{0}+\frac{\alpha _{1}}{\bar{S}-S-\alpha _{2}/(\alpha
_{3}+N)}=\alpha _{0}+\frac{\alpha _{1}(\alpha _{3}+N)}{(\bar{S}-S)(\alpha
_{3}+N)-\alpha _{2}}  \label{eq:MCMining}
\end{equation}%
illustrated in Figure \ref{fig:MiningTC}. For a given state of technical
knowledge $N$, the maximum fossil fuel resource that can be extracted is
given by $\bar{S}-\alpha _{2}/(\alpha _{3}+N)$. The terms $\alpha
_{0},\alpha _{1},\alpha _{2}$ and $\alpha _{3}$ in \eqref{eq:MCMining} are
parameters.

\begin{figure}[ht]
\centering \includegraphics[width=3.5in]{MiningTC.pdf}
\caption{Cost of energy from fossil fuels}
\label{fig:MiningTC}
\end{figure}

The absolute maximum fossil fuel available is given by $\bar{S}$, and this
is only available asymptotically as the stock of investment in new fossil
fuel technology $N\to\infty$. Even then, to exploit all the technically
available resources $\bar{S}$, would incur arbitrarily large costs.

For the later analysis, it also is useful to derive the partial derivatives
of the fossil fuel cost function $g(S,N)$. The fist partial derivatives are
given by 
\begin{equation}
\frac{\partial g}{\partial S}=\frac{\alpha_1(\alpha_3+N)^2}{[(\bar{S}%
-S)(\alpha_3+N)-\alpha_2]^2}>0  \label{eq:PartialgPartialS}
\end{equation}
and 
\begin{equation}
\frac{\partial g}{\partial N}=-\frac{\alpha_1\alpha_2}{[(\bar{S}%
-S)(\alpha_3+N)-\alpha_2]^2}<0  \label{eq:PartialgPartialN}
\end{equation}
so that increases in $S$ increase marginal cost, while improved technology
reduces the costs of providing fossil fuel energy. The second order partial
derivatives with respect to $S$ and $N$ are given by 
\begin{equation}
\frac{\partial^2 g}{\partial S^2}=\frac{2\alpha_1(\alpha_3+N)^3}{[(\bar{S}%
-S)(\alpha_3+N)-\alpha_2]^3}>0  \label{eq:Partial2gPartialS2}
\end{equation}
and 
\begin{equation}
\frac{\partial^2 g}{\partial N^2}=\frac{2\alpha_1\alpha_2(\bar{S}-S)}{[(\bar{%
S}-S)(\alpha_3+N)-\alpha_2]^3}>0  \label{eq:Partial2gPartialN2}
\end{equation}
In particular, this function implies that cumulative exploitation $S$
increases fossil fuel energy cost at an increasing rate, while investment in
fossil fuel technology decreases costs at a decreasing rate. In fact, we can
conclude from \eqref{eq:PartialgPartialN} that $\partial g/\partial
N\rightarrow 0$ as $N\rightarrow\infty$. The latter fact should imply that
eventually it becomes uneconomic to invest further in reducing the costs of
fossil fuel energy. Thus, fossil fuel resources will likely be abandoned
long before all known deposits are exhausted as rising costs make renewable
energy technologies more attractive.

Finally, the cross second partial derivative will be given by 
\begin{equation}
\frac{\partial^2 g}{\partial N\partial S}=-\frac{2\alpha_1\alpha_2(%
\alpha_3+N)}{[(\bar{S}-S)(\alpha_3+N)-\alpha_2]^3}<0
\label{eq:Partial2gPartialSN}
\end{equation}
Hence, investment in fossil fuel technology delays the increase in costs of
fossil fuel energy accompanying increased exploitation.

For energy to be productive on net, we need the value of output produced
from energy input to exceed the costs of producing that energy input. In
particular, if only fossil fuel is used to provide energy input, we must
have $1>g(S,N)$. The function \eqref{eq:MCMining} assumed above implies that
exhaustion of fossil fuel resources must eventually increase costs $g(S,N)$
so that this constraint is violated.

\subsection{Backstop Renewable Energy Technologies}

Motivated by the analysis that uses learning curves, we assume that the
marginal cost $p$ (measured in terms of goods) of the energy services
produced using the backstop technology declines as new knowledge is gained.
Following the literature examining learning-by-doing, we assume that
experience constructing capital using renewable energy input is the primary
factor in lowering the amount of such capital required to harvest the energy
needed to produce a given level of output. Even so, there is a limit, $%
\Gamma _{2}$, determined by physical constraints, below which $p$ cannot
fall. Explicitly, using $H$ to denote the stock of knowledge about backstop
energy production, and $\Gamma _{1}$ the initial value of $p$ (when $H=0$),
we assume: 
\begin{equation}
p=%
\begin{cases}
(\Gamma _{1}+H)^{-\alpha } & \text{ if $H\leq \Gamma _{2}{}^{-1/\alpha }-$}%
\Gamma _{1}, \\ 
\Gamma _{2} & \text{otherwise}%
\end{cases}
\label{eq:RenewCost}
\end{equation}%
for constant parameters $\Gamma _{1}$, $\Gamma _{2}$ and $\alpha $. We
assume that $\Gamma _{1}{}^{-\alpha }>g(0,0)$, so that renewable energy is
initially noncompetitive with fossil fuels.

We allow for technological progress to reduce the cost of renewable energy
through a learning curve. In our formulation, some direct R\&D expenditure $j$ can accelerate the accumulation of knowledge about the renewable technology:\footnote{%
Klaassen et. al. (2005) studied the impact of public R\&D and capacity
expansion on cost reducing innovation for wind turbine farms in Denmark,
Germany and the UK. They estimated a two-factor learning curve model that
allowed for both learning-by-doing and direct R\&D. They derive robust
estimates suggesting that direct R\&D is roughly twice as productive for
reducing costs as is learning-by-doing. They interpret their results as
enhancing the validity of the two-factor learning curve formulation.
Kouvaritakis et al. (2000) used a two-factor learning specification that
incorporates learning-by-doing effects as well as a relationship between
technology performance and R\&D expenditure.} 
\begin{equation}
\dot{H}=%
\begin{cases}
B(1+\psi j) & \text{ if $H\leq \Gamma _{2}^{-1/\alpha }-$}\Gamma _{1}, \\ 
0 & \text{otherwise}%
\end{cases}
\label{eq:Hdot}
\end{equation}%
In particular, once $H$ reaches its upper limit, further investment in the
technology would be worthless and we should have $j=0$. The parameter $\psi $
determines how investment in research enhances the accumulation of knowledge from experience.

As with fossil fuel, for the renewable backstop technology to be productive
on net, we require $p<1$. In effect, the renewable technology combines some
output (effectively, capital) with an exogenous energy source (for example,
sunlight, wind, waves or stored water) to produce more useful output than
has been used as an input.

\subsection{The Optimization Problem}

Goods are consumed, invested in $k$ or $H$, or used for producing fossil
fuel or backstop energy input. This leads to a resource constraint (in per capita
terms): 
\begin{equation}
c+i+j+n+g(S,N)R+pB=y  \label{eq:Budget}
\end{equation}

The objective function is maximized subject to the differential constraints %
\eqref{eq:Sdot}, \eqref{eq:Ndot}, \eqref{eq:kdot} and \eqref{eq:Hdot} with
initial conditions $S(0)=N(0)=0$, $k(0)=k_{0}>0$ and $H(0)=0$, the budget
constraint \eqref{eq:Budget}, the definitions of output \eqref{eq:ProdFn},
energy input \eqref{eq:NrgInput} and the evolution of the cost of the
backstop energy supply \eqref{eq:RenewCost}. The control variables are $%
c,i,j,R,n$ and $B$, while the state variables are $k,H,S$ and $N$. Denote
the corresponding co-state variables by $q,\eta ,\sigma $ and $\nu $. Let $%
\lambda $ be the Lagrange multiplier on the budget constraint. We also need
to allow for the possibility that either type of energy source is not used
and investment in cost reduction for the energy technology is zero. To that
end, let $\mu $ the multiplier on the constraint $j\geq 0$, $\omega $ the
multiplier on the constraint $n\geq 0$, $\xi $ the multiplier on the
constraint $R\geq 0$ and $\zeta $ the multiplier on the constraint $B\geq 0$%
. Finally, let $\chi $ be the multiplier on the constraint $H\leq \Gamma
_{2}{}^{-1/\alpha }-$$\Gamma _{1}$ on the accumulation of knowledge about
the renewable technology.

Define the current value Hamiltonian and thus Lagrangian by 
\begin{equation}
\begin{split}
& \mathcal{H}=\frac{c^{1-\gamma }}{1-\gamma }+\lambda \left[
Ak-c-i-j-n-g(S,N)R-(\Gamma _{1}+H)^{-\alpha }B\right] +\epsilon (R+B-Ak) \\
& +q(i-\delta k)+\eta B(1+\psi j)+\sigma QR+\nu n+\mu j+\omega n+\xi R+\zeta
B+\chi \lbrack \text{$\Gamma _{2}^{-1/\alpha }-$}\Gamma _{1}-H]
\end{split}
\label{eq:Hamiltonian}
\end{equation}

The first order conditions for a maximum with respect to the control
variables are: 
\begin{equation}
\frac{\partial \mathcal{H}}{\partial c}= c^{-\gamma} -\lambda = 0
\label{eq:FOCc}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial i}= -\lambda + q = 0  \label{eq:FOCi}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial j}=-\lambda + \eta\psi B + \mu = 0;
\mu j=0, \mu\ge 0, j \ge 0  \label{eq:FOCj}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial n}= -\lambda +\nu+\omega = 0, \omega
n=0, \omega\ge 0, n\ge 0  \label{eq:FOCn}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial R}= -\lambda g(S,N)+\epsilon+\sigma
Q+\xi = 0, \xi R=0, \xi\ge 0, R\ge 0  \label{eq:FOCR}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial B}= -\lambda
(\Gamma_1+H)^{-\alpha}+\epsilon+\eta(1+\psi j)+\zeta=0, \zeta B=0, \zeta\ge 0, B\ge 0
\label{eq:FOCB}
\end{equation}

The differential equations for the co-state variables are: 
\begin{equation}
\dot{q}=\beta q-\frac{\partial \mathcal{H}}{\partial k}=(\beta +\delta
)q-\lambda A+\epsilon A  \label{eq:qdot}
\end{equation}
\begin{equation}
\begin{split}
&\dot{\eta} =\beta \eta -\frac{\partial \mathcal{H}}{\partial H}=\beta \eta
-\lambda \alpha (\Gamma _{1}+H)^{-\alpha -1}B+\chi ; \\&
\chi \lbrack (\text{$\Gamma _{2}^{-1/\alpha }-$}\Gamma _{1}-H] =0,\chi \geq
0,H\leq \text{$\Gamma _{2}^{-1/\alpha }-$}\Gamma _{1}
\end{split}
\label{eq:etadot}
\end{equation}
\begin{equation}
\dot{\sigma}=\beta \sigma -\frac{\partial \mathcal{H}}{\partial S}=\beta
\sigma +\lambda \frac{\partial g}{\partial S}R  \label{eq:sigmadot}
\end{equation}%
\begin{equation}
\dot{\nu}=\beta \nu -\frac{\partial \mathcal{H}}{\partial N}=\beta \nu
+\lambda \frac{\partial g}{\partial N}R  \label{eq:nudot}
\end{equation}%
We also recover the budget constraint \eqref{eq:Budget} and the differential
equations for the state variables, \eqref{eq:kdot}, \eqref{eq:Hdot}, %
\eqref{eq:Sdot} and \eqref{eq:Ndot}.

\subsection{The Long Run Endogenous Growth Economy}

Since the costs of using fossil fuel must rise as resources are depleted,
ultimately energy is supplied using only the backstop renewable technology.
In the very long run, the cost of the renewable energy source will be
constant at $p=\Gamma _{2}$ and the stock of knowledge about renewable
energy production $H$ is no longer relevant. In this regime, the model
becomes a simple endogenous growth model with investment only in physical
capital. We retain the first order conditions \eqref{eq:FOCc}, %
\eqref{eq:FOCi} and \eqref{eq:FOCB}, the first co-state equation %
\eqref{eq:qdot}, the budget constraint \eqref{eq:Budget} and the
differential equation \eqref{eq:kdot} for the only remaining state variable $%
k$. However, \eqref{eq:FOCB} changes to simply $\epsilon =\lambda \Gamma
_{2} $. From \eqref{eq:FOCi} we will obtain $q=\lambda $ and hence $\dot{q}=%
\dot{\lambda}$, and the co-state equation \eqref{eq:qdot} becomes 
\begin{equation}
\dot{\lambda}=\left[ \beta +\delta -(1-\Gamma _{2})A\right] \lambda \equiv 
\bar{A}\lambda  \label{eq:term_lam_dot}
\end{equation}%
where $\bar{A}$ is a constant. If we are to have perpetual growth, we must
have $c\rightarrow \infty $ as $t\rightarrow \infty $, which from %
\eqref{eq:FOCc} will require $\lambda \rightarrow 0$ and hence $\bar{A}<0$,
that is\footnote{%
Note that \eqref{eq:Gam2Cond1} will require $A>(\beta +\delta )/(1-\bar{p}%
)>\beta +\delta $, which is the usual condition for perpetual growth in a
simple linear growth model.} 
\begin{equation}
\Gamma _{2}<1-\frac{\beta +\delta }{A}  \label{eq:Gam2Cond1}
\end{equation}%
Condition \eqref{eq:Gam2Cond1} has an intuitive interpretation. With $B=y$ and $p=\Gamma_2$, $A(1-\Gamma_2)$ equals output per unit of capital \textit{net} of the costs of supplying the backstop energy input. To obtain perpetual growth, this must exceed the cost of holding capital measured by the sum of the depreciation rate (the explicit cost) and the time discount rate (the implicit opportunity cost). Hereafter, we assume \eqref{eq:Gam2Cond1} to be valid. The solution to %
\eqref{eq:term_lam_dot} can be written 
\begin{equation}
\lambda =\bar{K}e^{\bar{A}t}  \label{eq:term_lam_sol}
\end{equation}%
for some constant $\bar{K}$ yet to be determined. Thus, in this final
regime, the budget constraint, the first order condition \eqref{eq:FOCc} for 
$c$ and \eqref{eq:term_lam_sol} imply 
\begin{equation}
\dot{k}=(\beta -\bar{A})k-\bar{K}^{-1/\gamma }e^{-\bar{A}t/\gamma }
\label{eq:TermBudget}
\end{equation}%
The integrating factor for the differential equation \eqref{eq:TermBudget}
is $e^{(\bar{A}-\beta )t}$, so the solution can be written 
\begin{equation}
k=C_{0}e^{(\beta -\bar{A})t}+\frac{\gamma \bar{K}^{-1/\gamma }e^{-\bar{A}%
t/\gamma }}{\beta \gamma -\bar{A}(\gamma -1)}  \label{eq:Term_k_gensol}
\end{equation}%
for another constant $C_{0}$. However, the transversality condition at
infinity requires 
\begin{equation}
\lim_{t\rightarrow \infty }e^{-\beta t}\lambda k=C_{0}+\lim_{t\rightarrow
\infty }\frac{\gamma \bar{K}^{-1/\gamma }e^{(-\bar{A}/\gamma +\bar{A}-\beta
)t}}{\beta \gamma -\bar{A}(\gamma -1)}=0  \label{eq:TVC}
\end{equation}%
Equation \eqref{eq:TVC} in turn requires 
\begin{equation}
C_{0}=0\,\text{ and }\,\bar{A}(\gamma -1)<\beta \gamma  \label{eq:TVCresult}
\end{equation}%
Note that since $\bar{A}<0$ the inequality in \eqref{eq:TVCresult} will be satisfied if $%
\gamma >1$, while if $0<\gamma <1$, it will require 
\begin{equation}
\Gamma _{2}>1-\frac{\beta /(1-\gamma )+\delta }{A}  \label{eq:Gam2Cond2}
\end{equation}%
Thus, for $\gamma <1$, the validity of \eqref{eq:Gam2Cond1} and %
\eqref{eq:Gam2Cond2} together require

\begin{equation}
1-\frac{\beta +\delta }{A}>\Gamma _{2}>1-\frac{\beta /(1-\gamma )+\delta }{A}
\label{eq:pbarConstr}
\end{equation}

In summary, we conclude that the value of $k$ in the final endogenous growth
economy will be given by 
\begin{equation}
k = \frac{\gamma \bar{K}^{-1/\gamma}e^{-\bar{A}t/\gamma}}{\beta\gamma-\bar{A}%
(\gamma-1)}  \label{eq:Term_k_sol}
\end{equation}
with $\lambda$ given by \eqref{eq:term_lam_sol} and $\bar{K}$ is a constant
yet to be determined.

For periods prior to the terminal endogenous growth regime just analyzed, note first that we cannot have $j>0$ and $B=0$.  This follows from \eqref{eq:FOCj}, since if $B=0$, $\mu=\lambda>0$ which implies $j=0$.  For empirically relevant parameter values, however, we can have a short interval of time where $B>0$ and $j=0$. Since $B>0$ in this regime, learning by doing implies that the cost of renewable energy will decline.

Since the energy services of fossil fuels and the backstop renewable technology are perfect substitutes, the renewable technology will not be competitive with fossil fuels until the shadow price of energy in the fossil fuel regime equals the shadow price in the renewable backstop regime. When only fossil fuel is used, we assume that the productivity of investing in fossil fuel technology is high enough to sustain investments right up until the time the economy transitions to renewable energy. Although investments moderate the increase in fossil fuel costs, eventually depletion ensures that the shadow price of energy derived from fossil fuels rises to equal the initially higher cost of energy from renewable sources. At that point, the economy switches to use only renewable energy and all investment in, and use of, fossil fuel technologies ceases. We therefore conclude that the economy will pass through the regimes illustrated in the time line in Figure \ref{fig:Regimes}.

\begin{figure}[ht]
\centering \includegraphics[width=6.5in]{FourRegime.pdf}
\caption{Regimes of energy use and investment}
\label{fig:Regimes}
\end{figure}

\subsection{The Initial Fossil Fuel Economy}

It is useful to consider next the regime where $R>0$. Then \eqref{eq:FOCR} implies $\xi =0$ and the shadow price of energy
will be 
\begin{equation}
\epsilon =\lambda g(S,N)-\sigma Q  \label{eq:NrgP_fossregime}
\end{equation}%
Since an increase in $S$ raises the costs of fossil fuel, the co-state
variable $\sigma $ will be negative\footnote{%
Recall that if we use $V$ to denote the maximized value of the objective subject to the
constraints, the co-state variable $\sigma $ will equal the partial
derivative of $V$ with respect to the corresponding state variable, $S$.}
while fossil fuels are used as an energy source. It then follows from %
\eqref{eq:NrgP_fossregime} that the shadow price of energy $\epsilon $ is
unambiguously positive.

We also assume parameter values are chosen so that investment in fossil fuel technology is productive, that is, $n>0$. Then \eqref{eq:FOCn} implies $\omega =0$ and hence $\nu =\lambda $. But then $\dot{\nu}=\dot{\lambda}$ and \eqref{eq:nudot} implies
\begin{equation}
\dot{\lambda}=\beta\lambda+\lambda\frac{\partial g}{\partial N}R
\label{eq:lamdotfoss}
\end{equation}
If we also have $i>0$, \eqref{eq:FOCi} will imply $\lambda=q$ and from \eqref{eq:qdot} and \eqref{eq:NrgP_fossregime}, we will also have $\dot{\lambda}=(\beta+\delta+g(S,N)A-A)\lambda-\sigma QA$. Using \eqref{eq:lamdotfoss} we then conclude
\begin{equation}
\left[ \delta +g(S,N)A-\frac{\partial g}{\partial N}R-A\right] \lambda
=\sigma QA  \label{eq:qdoteqnudot}
\end{equation}%
Note that since $\sigma <0$ and $\lambda =c^{-\gamma }>0$, a necessary
condition for \eqref{eq:qdoteqnudot} to hold is that 
\begin{equation}
\delta +g(S,N)A-\frac{\partial g}{\partial N}R<A  \label{eq:FossNC}
\end{equation}%
Since we have assumed, however, that $g(S,N)$ eventually increases above $1$
as $S$ grows, and $\partial g/\partial N<0$, constraint \eqref{eq:FossNC}
must eventually be violated and the economy will not use fossil fuels
forever.

Substituting $R=Ak$ into \eqref{eq:qdoteqnudot}, we obtain an equation relating $N$ and $k$. When there is active investment in two types of capital (here $k$ and $N$), the investment has to maintain a relationship between the two stocks. Differentiating the resulting expression with respect to time, substituting for $\dot{N},\dot{\lambda}/\lambda =\dot{\nu}/\nu ,\dot{S},\dot{\sigma}$ and $\dot{Q}=\pi Q$ (since the exogenous growth rate of $Q$ is $\pi $), and using \eqref{eq:qdoteqnudot}, we obtain a condition relating the two types of investments ($i$ and $n$) in the initial fossil fuel economy: 
\begin{equation}
\lambda \left[ \frac{\partial g}{\partial N}(n+\delta k+\frac{\sigma QAk}{%
\lambda }-i)-\frac{\partial ^{2}g}{\partial S\partial N}QAk^{2}-\frac{%
\partial ^{2}g}{\partial N^{2}}nk\right] =\sigma \pi Q  \label{eq:i_n}
\end{equation}

We obtain a second relationship from the budget constraint. Specifically, using the result that $j=0$ if $B=0$, the first order condition \eqref{eq:FOCc}
for $c$, the production function \eqref{eq:ProdFn}, the energy input demand
requirement \eqref{eq:NrgInput} the budget constraint \eqref{eq:Budget} implies: 
\begin{equation}
i=Ak[1-g(S,N)]-\lambda ^{-1/\gamma }-n  \label{eq:i_fossregime}
\end{equation}%
Substituting \eqref{eq:i_fossregime} into \eqref{eq:i_n}, we then obtain an
equation to be solved for energy technology investment $n$ in the fossil
fuel regime: 
\begin{equation}
\begin{split}
& n\lambda\left( \frac{\partial ^{2}g}{\partial N^{2}}k-2\frac{\partial g}{%
\partial N}\right) =\\&\lambda \left[ \frac{\partial g}{\partial N}[k(\delta
+g(S,N)A-A+\frac{\sigma QA}{\lambda })+\lambda ^{-1/\gamma }]-\frac{\partial
^{2}g}{\partial S\partial N}QAk^{2}\right] -\sigma \pi Q  
\end{split}
\label{eq:n_soln}
\end{equation}%
Since $\partial g/\partial N<0$ and $\partial ^{2}g/\partial N^{2}>0$, the
coefficient of $n$ on the left hand side of \eqref{eq:n_soln} is positive.
From the budget constraint \eqref{eq:i_fossregime}, $\delta
k+Ak(g-1)+\lambda ^{-1/\gamma }=\delta k-i-n\le\delta k-n$. Then if
\begin{equation}
-\frac{\partial^{2}g}{\partial S\partial N}QAk^{2}+\frac{\partial g}{\partial N}(\delta+\frac{\sigma QA}{\lambda }) k-\sigma \pi Q>0
\label{eq:nposcond}
\end{equation}
we can conclude that $n>0$ as hypothesized.\footnote{Since $\partial^{2}g/\partial S\partial N<0$ and $\sigma<0$, the quadratic in $k$ in \eqref{eq:nposcond} has a positive second derivative and positive intercept, so even if $\delta+\sigma QA/\lambda>0$, so the roots are both positive, we conclude that \eqref{eq:nposcond} must hold for large $k$. For small values of $k$, we are likely to have $\dot{k}=i-\delta k>0$, in which case the right hand side of \eqref{eq:n_soln} is guaranteed to be positive.} Using the solution for $n$ and the current values of the state and co-state variables, \eqref{eq:i_fossregime} can be solved for $i$.

In summary, we conclude that the initial period of fossil fuel use with both $i>0$ and $n>0$ produces five differential equations
for $k$, $S$, $N$, $\sigma $, and $\lambda $: 
\begin{equation}
\dot{k}=i-\delta k 
\label{eq:Reg1_keq}
\end{equation}
\begin{equation}
\dot{S}=QAk 
\label{eq:Reg1_Seq}
\end{equation}
\begin{equation}
\dot{N}=n 
\label{eq:Reg1_Neq}
\end{equation}
\begin{equation}
\dot{\sigma}=\beta \sigma +\lambda \frac{\partial g}{\partial S}Ak
\label{eq:Reg1_sigeq}
\end{equation}
\begin{equation}
\dot{\lambda}=\lambda(\beta+\delta+(g(S,N)-1)A)-\sigma QA
\label{eq:Reg1_lameq}
\end{equation}
together with the exogenous population growth $Q=Q_{0}e^{\pi t}$.

\subsection{The Intermediate Economy with Renewables and Technological
Progress}

We next consider the regimes where $B=Ak>0,j\ge 0$ and $H<\Gamma _{2}^{-1/\alpha}- \Gamma _{1}$. For $B>0$, \eqref{eq:FOCB} implies $\zeta =0$, while $H<\Gamma _{2}^{-1/\alpha}- \Gamma _{1}$ and \eqref{eq:etadot} imply $\chi =0$. Considering first the majority of this regime where $j>0$,  \eqref{eq:FOCj} implies $\mu =0$, and from \eqref{eq:FOCi} and \eqref{eq:FOCj},  $q=\lambda =\eta\psi Ak$. Thus, when $j>0$ the shadow price of energy becomes

\begin{equation}
\epsilon =\lambda (\Gamma _{1}+H)^{-\alpha }-\frac{\lambda(1+ \psi j)}{\psi Ak}
\label{eq:BstopNrgP}
\end{equation}%
Substituting \eqref{eq:BstopNrgP} into \eqref{eq:qdot} and noting that $%
q=\lambda $ implies $\dot{q}=\dot{\lambda}$, we obtain 
\begin{equation}
\frac{\dot{\lambda}}{\lambda }=\beta +\delta -A(1-(\Gamma _{1}+H)^{-\alpha
})-\frac{1}{\psi k}-\frac{j}{k}  \label{eq:lamdotreg2}
\end{equation}%
From \eqref{eq:etadot} with $\lambda =\eta\psi Ak$ and $B=Ak$, and
using $\dot{k}=i-\delta k$, we obtain 
\begin{equation}
\frac{\dot{\lambda}}{\lambda }=\beta-\delta -\alpha (\Gamma _{1}+H)^{-\alpha
-1}\psi (Ak)^2+\frac{i}{k}
\label{eq:etadotreg2}
\end{equation}%
Equating \eqref{eq:lamdotreg2} and \eqref{eq:etadotreg2}, we obtain an
expression for total investment, $i+j$, as a function of $k$
and $H$ 
\begin{equation}
i+j=2\delta k-\frac{1}{\psi}-Ak(1-(\Gamma _{1}+H)^{-\alpha})+\alpha \psi A^2k^{3}(\Gamma _{1}+H)^{-\alpha -1}  \label{eq:Invreg2}
\end{equation}%
The budget constraint and the first order condition \eqref{eq:FOCc} for $c$
then provide a second equation for $i+j$: 
\begin{equation}
i+j=Ak(1-(\Gamma _{1}+H)^{-\alpha })-\lambda ^{-1/\gamma }
\label{eq:budgetreg2}
\end{equation}%
Substituting \eqref{eq:budgetreg2} into \eqref{eq:Invreg2}, we obtain an
equation relating $H$ and $k$: 
\begin{equation}
\alpha \psi A^2k^{3}(\Gamma _{1}+H)^{-\alpha -1}+2k[\delta-A(1-(\Gamma _{1}+H)^{-\alpha })]+\lambda ^{-1/\gamma }-\frac{1}{\psi}=0
\label{eq:Hkreg2}
\end{equation}%
Once again, when there is active investment in two types of capital (here $k$ and $H$), the investment has to maintain a relationship between the two stocks.\footnote{It can be shown that \eqref{eq:Hkreg2} has a unique real solution for $k$ in terms of $H$ and the parameters.}

Differentiating \eqref{eq:Hkreg2} with respect to $t$, and substituting $%
\dot{k}=i-\delta k$, $\dot{H}=Ak(1+\psi j)$ and for $\dot{\lambda}/\lambda $
using \eqref{eq:lamdotreg2} we obtain a second relationship between $i$ and $%
j$ and the current values of $k,H$ and $\lambda $:

\begin{equation}
\begin{split}
& \left[ 2[\delta -A(1-(\Gamma _{1}+H)^{-\alpha })]+3\alpha\psi (Ak)^2(\Gamma_{1}+H)^{-\alpha -1}\right] i+ \\
& \left[ \frac{\lambda ^{-1/\gamma }}{\gamma k}-\alpha\psi (Ak)^2(\Gamma _{1}+H)^{-\alpha -1}[2+(1+\alpha)\psi Ak^2(\Gamma _{1}+H)^{-1}]\right] j \\
& =\left[ 2[\delta -A(1-(\Gamma _{1}+H)^{-\alpha })]+3\alpha\psi (Ak)^2(\Gamma_{1}+H)^{-\alpha -1}\right] \delta k \\
&+\alpha (Ak)^2(\Gamma _{1}+H)^{-\alpha -1}[2+(1+\alpha)\psi Ak^2(\Gamma _{1}+H)^{-1}]\\
& +\frac{\lambda ^{-1/\gamma }}{\gamma }[\beta +\delta -A(1-(\Gamma_{1}+H)^{-\alpha })-\frac{1}{\psi k}]
\end{split}
\label{eq:ijrelreg2}
\end{equation}%
The two equations \eqref{eq:budgetreg2} and \eqref{eq:ijrelreg2} can then be
solved for $i$ and $j$ given current values for $k,H$ and $\lambda $. Using
the solutions for the investment levels, we can then obtain the differential
equations governing the evolution of $k,H$ and $\lambda $ in region 2,
which, for convenience are summarized below: 
\begin{gather}
\dot{k}=i-\delta k  \label{eq:Reg2_keq} \\
\dot{H}=Ak(1+\psi j)  \label{eq:Reg2_Heq}\\
\dot{\lambda}=\lambda \left[ \beta +\delta -A(1-(\Gamma _{1}+H)^{-\alpha })-\frac{1+\psi j}{\psi k}\right]  \label{eq:Reg2_lameq}
\end{gather}

Now consider the beginning of the renewable energy regime where $j=0$. Here we will have from \eqref{eq:FOCj} that $\mu=\lambda-\eta\psi B\ge 0$ with $\lambda=\eta\psi B$ at the upper boundary where the constraint on $j$ is just binding. In the interior of this region where $n=j=0=R$, the budget constraint \eqref{eq:Budget} will imply $i=Ak(1-p)-c$ with $p=(\Gamma_1+H)^{-\alpha}$ and $c=\lambda^{-1/\gamma}$. Also, the shadow price of energy obtained from \eqref{eq:FOCB} will now be given by $\epsilon=\lambda p-\eta$ and we retain $q=\lambda$, but we will no longer have $\lambda=\eta\psi Ak$. The differential equations governing the evolution of $k,H,\lambda$ and $\eta $ now become: 
\begin{gather}
\dot{k}=Ak[1-(\Gamma_1+H)^{-\alpha}]-\lambda^{-1/\gamma}-\delta k  \label{eq:Reg2a_keq} \\
\dot{H}=Ak  \label{eq:Reg2a_Heq} \\
\dot{\lambda}=\lambda \left[ \beta +\delta -A(1-(\Gamma _{1}+H)^{-\alpha })\right]-\eta A  \label{eq:Reg2a_lameq}\\
\dot{\eta}=\beta\eta-\lambda\alpha(\Gamma _{1}+H)^{-\alpha-1}Ak \label{eq:Reg2a_etaeq}
\end{gather}

\subsection{Boundary conditions}

In the numerical analysis, the economy begins with known values of the state variables $k(0),S(0)$ and $N(0)$ at $t=0$. However, the initial values of the co-state variables $\lambda(0)$ and $\sigma(0)$ are unknown. Similarly, the initial value of the co-state variable $\eta(T_0)$ at $T_0$ is unknown. These all have to be guessed and the model solved forward. The values of the co-state variables at the transition times are then compared with their target values and the guesses are modified until all the targets are attained to the desired numerical accuracy. In this section, we discuss what the target values ought to be.

First, note that at $T_0$, $H=j=0=\sigma$. Using \eqref{eq:BstopNrgP} and \eqref{eq:NrgP_fossregime}, the fact that and the requirement that the shadow price of energy has to be continuous across the region boundaries then implies 
\begin{equation}
\Gamma _{1}^{-\alpha }-\frac{\eta}{\lambda}=\frac{\epsilon}{\lambda}= g(S,N) 
\label{eq:T_0NrgP}
\end{equation}
For a given value of $\eta(T_0)$, \eqref{eq:T_0NrgP} would then determine a value of $T_0$ and corresponding values of $k(T_0),S(T_0),N(T_0),\sigma(T_0)$ and $\lambda(T_0)$  using the differential equations \eqref{eq:Reg1_keq}--\eqref{eq:Reg1_lameq}. The calculated value of $\sigma(T_0)$ would then need to be compared to its target value of 0.

The calculated values for $k(T_0)$ and $\lambda(T_0)$ together with $H=0$ and the guessed value of  $\eta(T_0)$ will then provide starting values for the differential equations \eqref{eq:Reg2a_keq}--\eqref{eq:Reg2a_etaeq} in the next regime.\footnote{It may also be worth noting that a number of control variables will not be continuous across the $t=T_0$ boundary. To begin with, $R$ will jump from equalling $Ak>0$ right up until $T_0$ to a value of zero at $T_0$ and beyond. Correspondingly, $B$ will jump from zero before $T_0$ to $Ak>0$ from $T_0$ on. In addition, $n$ will jump from being strictly positive as $t\to T_0$ to being zero at $T_0$. Conversely, $j$ will jump from zero for $t<T_0$ to being positive at $T_0$.} As already noted above, the upper boundary $T_1$ of this region will occur where $\lambda=\eta\psi Ak$.

Once $T_1$ has been reached, the differential equations change to \eqref{eq:Reg2_keq}--\eqref{eq:Reg2_lameq}. Again, the values of $k,H$ and $\lambda$ will be continuous across the $T_1$ boundary. We also require that the initial calculated value for $j$ in the third regime, using \eqref{eq:budgetreg2} and \eqref{eq:ijrelreg2}, equal its target value of 0.  The upper boundary of this third region, $T_2$, will occur where $p=(\Gamma_{1}+H)^{-\alpha }=\Gamma _{2}$, which will determine the value of $H$ at $T_{2}$, namely $H=\Gamma_{2}^{-1/\alpha }-\Gamma _{1}$. We will also know the values of $k$ and $\lambda $ at $T_{2}$ (up to the unknown constant $\bar{K}$) since they must be continuous across the boundary and therefore must equal \eqref{eq:Term_k_sol} and \eqref{eq:term_lam_sol} respectively. One of these equations, say \eqref{eq:term_lam_sol}, can be used to solve for $\bar{K}$ and  then \eqref{eq:Term_k_sol} will provide a third target value.

Note that in total we have three targets $\sigma(T_0)=0,j(T_1)=0$ and $k(T_2)$ equals the corresponding calculated value implied by $\lambda(T_2)$, that can be used to determine appropriate initial values for the three variables $\lambda(0),\sigma(0)$ and $\eta(T_0)$. In practice, we guess values for the latter and iterate until the targets are attained.

\subsection{Calibration}

In order to quantitatively evaluate different policy scenarios, we first need to calibrate the theoretical model. This involves assigning numerical values to certain parameters in a way that make the model consistent with observations from the actual world economy. By definition, we start the economy with $S=N=H=0$ and with $Q=Q_{0}$. For convenience, we take the current population $Q_{0}=1$ and effectively measure future population as multiples of the current level. We will assume that the population growth rate is 1\%.\footnote{This is consistent with a simple extrapolation of recent world growth rates reported by the Food And Agriculture Organization of the United Nations, \textsf{http://faostat.fao.org/site/550/default.aspx}}

In line with standard assumptions made to calibrate growth models, we assume a time discount factor $\beta =0.05$. From previous analyses, we would expect the coefficient of relative risk aversion $\gamma$ to lie between 1 and 10, but there is no strong consensus on what the value should be. As we explain in more detail below, we will allow $\gamma$ to adjust to ensure we match the initial share of consumption in GDP.

To calibrate values for the initial production, capital and energy quantities we used data from the \textit{Energy Information Administration} (EIA),\footnote{International data is available at \textsf{http://www.eia.doe.gov/emeu/international/contents.html}} the \textit{Survey of Energy Resources 2007} produced by the \textit{World Energy Council},\footnote{This is available at \textsf{http://www.worldenergy.org/publications/survey\_of\_energy\_resources\_2007/default.asp} The data are estimates as of the end of 2005.} and \textit{The GTAP 7 Data Base} produced by the \textit{Center for Global Trade Analysis} in the Department of Agricultural Economics, Purdue University.\footnote{Information on this can be found at \textsf{https://www.gtap.agecon.purdue.edu/databases/v7/default.asp} The GTAP 7 data base pertains to data for 2004.} The last mentioned data source is useful for our purposes because it provides a consistent set of international
accounts that also take account of energy flows.

One of the first issues we need to address is that national accounts include government spending in GDP, which does not appear in the model.\footnote{Note that in the GTAP data base, aggregate world exports equal aggregate world imports so world GDP equals consumption plus investment plus government expenditure.} We therefore subtracted government spending from the GDP measures before calibrating the remaining variables. Conceptually, this would be correct if the utility obtained from government spending were additively separable from the utility obtained from private consumption and government spending was financed by lump sum taxes. In practice, neither of these assumptions is valid and government activity (apart from energy taxes or subsidies, which will be considered explicitly later) would affect the equilibrium of the model.

After excluding government, the investment share of private sector expenditure is 0.2575. Effectively defining units so that aggregate output is 1, we therefore identify 0.2575 as the sum $i+n$ at $t=0$. We would expect most of this to be investment in capital used to produce output rather than fossil fuel exploration and development.

Converting the GTAP data base estimates of the total capital stock to units of GDP, we obtain the initial condition $k(0)=3.2759$. We also use the GTAP depreciation rate on capital of 4\%. Also, if we choose units so that output equals 1, the parameter $A$ would equal the ratio of output to capital, that is, $A\approx 0.3053$.

From the budget constraint, the difference between total output and the sum of the investments, namely 0.7425 would equal consumption plus the current costs $gR$ of supplying fossil fuels. We separated these two components using sectoral data from the GTAP data base. Specifically, we classified ``energy expenditure'' as combined spending on the primary fuels coal, oil and natural gas and the energy commodity transformation sectors of refining, chemicals, electricity generation and natural gas distribution. The current cost of fossil energy was then set equal to the expenditure on these sectors that was classified as consumption rather than investment. This produced a value for $gR=0.0558$.

Subtracting the initial value for $gR$ from 0.7425 we obtain the initial value of $c(0)=0.6867$. As noted above, the normal method of solving the the optimal control problem would involve specifying values for the parameters and the state variables and then solving for values of the co-state variables that allow us to hit required terminal values. The value for $c(0)$ would then follow from the first order condition $\lambda(0)=c(0)^{-\gamma}$. To obtain a particular value for $c(0)$ we then need to free up an additional parameter. As already noted above, we will introduce $c(0)$ as a new target and adjust the value of $\gamma$ as $\lambda(0)$ changes to ensure that  $\lambda(0)=c(0)^{-\gamma}$ always remains valid.

After we set the initial values of $S$ and $N$ to zero, the initial value for $gR$ also would imply 
\begin{equation}
\frac{0.0558}{R}=\alpha _{0}+\frac{\alpha _{1}}{\bar{S}-\alpha _{2}/\alpha_{3}}
\label{eq:Init_g}
\end{equation}
We can obtain a value for total fossil fuel production, $R$, from the EIA web site. It gives world wide production of oil in 2004 of 175.948 quads (where one quad equals $10^{15}$ BTU), of natural gas 100.141 quads and of coal 116.6 quads. Summing these gives a total of 392.689 quads. We then choose energy units so that the initial value of $R=1$.

To obtain an estimate of total fossil fuel resources $\bar{S}$ in the same units, we begin with the proved and estimated additional resources in place from the World Energy Council. The millions of tonnes of coal, millions of barrels of oil, extra heavy oil, natural bitumen and oil shale and trillions of cubic feet of natural gas given in that publication were converted to quads using conversion factors available at the EIA. The result is 115.2 quintillion BTU, or almost 300 times the annual worldwide production of fossil fuels in 2004. These resources are nevertheless relatively small compared to estimates of the volume of methane hydrates that may be available. Although experiments have been conducted to test methods of exploiting methane hydrates, a commercially viable process is yet to be demonstrated. Partly as a result, resource estimates vary widely. According to the National Energy Technology Laboratory (NETL),\footnote{\textsf{http://www.netl.doe.gov/technologies/oil-gas/FutureSupply/MethaneHydrates/about-hydrates/estimates.htm}} the United States Geological Survey (USGS) has estimated potential resources of about 200,000 trillion cubic feet in the United States alone. According to Timothy Collett of the USGS,\footnote{\textsf{http://www.netl.doe.gov/kmd/cds/disk10/collett.pdf}} current estimates of the worldwide resource in place are about 700,000 trillion cubic feet of methane. Using the latter figure, this would be equivalent to 719.6 quintillion BTU. Adding this to the previous total of oil, natural gas and coal resources yields a value for $\bar{S}=834.8$ quintillion BTU or around 2125.8986 in terms of the energy units defined so that $R=1$.

We still need to specify values for the $\alpha_i$ parameters in the $g$ function. Equation \eqref{eq:Init_g} with $R\equiv 1$ will give us one equation in four unknowns. Noting that we can interpret $\bar{S}-\alpha_2/\alpha_3$ as the initial level of fossil fuel extraction $S$ at which marginal costs of extraction $g(S,0)$ would become unbounded, we associate  $\bar{S}-\alpha_2/\alpha_3$ with current proved and connected reserves of fossil fuel.\footnote{Note that current official reserves are not the relevant measure since many of these are not connected and thus are unavailable for production without further investment, denoted $n$ in the model.} A recent report from Cambridge Energy Research Associates (CERA, 2009),\footnote{``The Future of Global Oil Supply: Understanding the Building Blocks,'' Special Report by Peter Jackson, Senior Director, IHS Cambridge Energy Research Associates, Cambridge, MA.} for example, gives weighted average decline rates for oil production from existing fields of around 4.5\% per year. They also note that this figure is dominated by a small number of ``giant'' fields and that, ``the average decline rate for fields that were actually in the decline phase was 7.5\%, but this number falls to 6.1\% when the numbers are production weighted.'' Hence, we shall use 6\% as a decline rate for oil fields. If we use United States production and reserve figures as a guide, we find that natural gas decline rates are closer to 8\% per year but coal mine decline rates are closer to 6\% per year. In accordance with these figures, we assume the ratio of fossil fuel production to proved and connected reserves equals the share weighted average of these figures, namely $(175.948\ast 0.06+100.141\ast 0.08+116.6\ast 0.06)/392.689=0.0651$. Thus, in terms of the energy units defined so that $R=1$, the initial target value of $\bar{S} -\alpha_2/\alpha_3$ would equal 1/0.0651=15.361. Using the previously calculated value for $\bar{S}$, this leads to $\alpha_2/\alpha_3=2110.538$.

\begin{figure}[ht]
\centering \includegraphics[width=5in]{MiningMCcurves.pdf}
\caption{$g(S,N)$ for $N=0.3$ and different values of $g_S$ and $g_N$}
\label{fig:MiningMCcurves}
\end{figure}

We can obtain two more equations by specifying the partial derivatives of $g$ at $t=0$. Using GTAP data on capital shares by sector, we estimate that around 6.5\% of annual investment occurs in the oil, natural gas, coal, electricity,  and gas distribution sectors.\footnote{Since we have defined $R$ to be energy services input, investments in energy efficiency in addition to mining increase the effective supply of fossil fuels. Hence, we include investments in the energy transformation sectors. While some of these would not increase energy efficiency, some investments in the transportation and manufacturing sectors that have not been included would be aimed at raising energy efficiency.} We noted above that in the GTAP data, total investment $i+n=0.2575$, implying that $n\approx 0.0167$ in private sector output units. We assume that this level of investment at $t=0$ is sufficient to replace mined resources and allow for growth in total annual fossil fuel production equivalent to the average annual growth over 2004-08 of around 2.36\%.\footnote{These calculations are again based on data from the EIA.} Specifically, with $\alpha_2/\alpha_3=2110.538$, we assume that $\alpha_2/(\alpha_3+0.0167)=2109.195$, which implies $\alpha_3\approx 25.852$. The previously calculated value for $\alpha_2/\alpha_3$ then implies $\alpha_2 \approx 54561.15$. Given values for $\alpha_2$ and $\alpha_3$, the ratio $g_N/g_S$ then also is determined, but the individual values of $g_S$ and $g_N$ can still vary. As they do, $\alpha_0$ and $\alpha_1$ also will vary. Figure \ref{fig:MiningMCcurves} illustrates the curves at $t=0$ for values of $g_S$ ranging from 0.0004 (the closest to a right angled shape) to 0.001 (the furthest from a right angled shape). We assumed $g_S=0.001$ for the following calculations.

Turning next to the learning curve \eqref{eq:RenewCost}, the literature
provides a range of estimates for $\alpha $. An online calculator provided
by NASA\footnote{Available at \textsf{http://cost.jsc.nasa.gov/learn.html}} gives a range of
learning percentages between 5 and 20\% depending on the industry. A
learning percentage of $x$, which corresponds to a value of $\alpha
=-ln(1-x)/ln(2)$, has the interpretation that a doubling of the experience
measure will lead to a cost reduction of $x$\%. Thus, $x=0.2$ is equivalent
to $\alpha =0.322$ while $x=.05$ corresponds to $\alpha =0.074$. In a study
of wind turbines, Coulomb and Neuhoff (2006)\footnote{Louis Coulomb and Karsten Neuhoff, \textquotedblleft Learning Curves and
Changing Product Attributes: the Case of Wind Turbines\textquotedblright ,
University of Cambridge: Electricity Policy Research Group, Working Paper
EPRG0601.} found values of $\alpha $ of 0.158 and 0.197. In a 1998 paper, Gr\"{u}bler and Messner\footnote{Arnulf Gr\"{u}bler and Sabine Messner, \textquotedblleft Technological
change and the timing of mitigation measures\textquotedblright , \textit{Energy Economics} 20, 1998, 495--512} found a value for $\alpha =.36$ using
data on solar panels. In a 2008 paper in \textit{The Energy Journal}, van
Bentham et. al.\footnote{\textquotedblleft Learning-by-doing and the optimal solar policy in
California,\textquotedblright\ Arthur van Benthem, Kenneth Gillingham and
James Sweeney, 29(3) 2008, 131-152} report several studies finding a
learning percentage of around 20\% ($\alpha =0.322$) for solar panels. For
our base case, we will take $\alpha =0.37$.

The other parameter affecting the incentive to invest in renewable energy
sources is the initial value $\Gamma _{1}^{-\alpha }$ of the cost of using
renewable energy as the primary energy source. Using a document available
from the Energy Information Administration (EIA)\footnote{\textit{%
Assumptions to the Annual Energy Outlook, 2009}, \textquotedblleft
Electricity Market Module,\textquotedblright\ Table 8.2, available at 
\textsf{http://www.eia.doe.gov/oiaf/aeo/assumption/pdf/electricity.pdf%
\#page=3}} the cost of new onshore wind capacity is about double the cost of
combined cycle gas turbines (CCGT), while offshore wind is around four times
as expensive, solar thermal more than five times as expensive and solar
photovoltaic more than six times as expensive. However, these costs do not
take account of the lower average capacity factor of intermittent sources
such as wind or solar. The same document gives a fixed O\&M cost of onshore
wind that is around two and a half times the corresponding fixed O\&M for
CCGT, although the latter also has fuel costs. The corresponding ratio is
around 7 for offshore wind, while fixed O\&M for solar photovoltaic are
similar to the fixed O\&M for CCGT. As a rough approximation, we will assume 
$\Gamma _{1}^{-\alpha }$ is around 4 times the initial
value of $g$. In accordance with the EIA assumptions, we also assume that,
in the long run, the renewable technologies can experience a five-fold
reduction in costs, so $\Gamma _{2}=\Gamma _{1}^{-\alpha }/5$. This would result in an energy
cost that is below the current cost of fossil fuel technologies.

Finally, we need to specify a value for $\psi $, the relative effectiveness
of direct investment in research versus learning by doing in accumulating
knowledge about new energy technologies. Klaassen et. al. (2005)\footnote{%
Klaassen, Ger, Asami Miketa, Katarina Larsen and Thomas Sundqvist,
\textquotedblleft The impact of R\&D on innovation for wind energy in
Denmark, Germany and the United Kingdom,\textquotedblright\ \textit{%
Ecological Economics}, 54 (2005) 227--240} estimated a model that allowed
for both learning-by-doing and direct R\&D. Although they assume the capital
cost is multiplicative in total R\&D and cumulative capacity, while we assume the 
\textit{change} in knowledge is multiplicative in new R\&D and cumulative
capacity, we can take their parameter estimates as a guide. They find direct
R\&D is roughly twice as productive for reducing costs as is
learning-by-doing.\footnote{%
Of course, the learning-by-doing has the advantage that it directly
contributes to output at the same time it is adding to knowledge.}
Consequently, we assume that $\psi =2$.

The results from the calibrated version of our model economy are summarized
below. Absent any government intervention in the economy, the transition to
a renewable energy regime occurs after $T_0=38.936$ years. Renewable energy is then used for a little less than 15 years (until $T_1=53.5668$) before direct R\&D expenditure $j$ becomes worthwhile. The renewable technology then reaches its ultimate frontier around 13 years later at $T_2=67.0686$ years.

\begin{figure}[ht]
\centering \includegraphics[width=6.25in]{FossilRegime.pdf}
\caption{Fossil fuel regime without taxes or subsidies}
\label{fig:Regime1NoTaxes}
\end{figure}

Figure~\ref{fig:Regime1NoTaxes} shows the behavior of the main
variables in the economy during the initial regime. Fossil fuel use leads to
growth in consumption, as well as in the economy's capital stock. However,
increasing investment in the development of mining technologies is necessary
to meet demand as the economy grows. Towards the end of regime 1, the costs
associated with increase fossil fuel use are large in real terms. This paves the way for the
transition to the renewable energy regime.

\begin{figure}[ht]
\centering \includegraphics[width=6.5in]{RenewableRegime.pdf}
\caption{Renewable regime without taxes or subsidies}
\label{fig:Regime2NoTaxes}
\end{figure}

Figure~\ref{fig:Regime2NoTaxes} shows the behavior of the main
variables in the first renewable energy regime where technological progress continues to reduce renewable energy costs. Here, economic production is fueled through the use of renewable energy. Direct investment in
renewable energy increases over time. Together with learning-by-doing, this
leads to the accumulation of technical knowledge that is necessary for a more
efficient use of this technology. Consumption and the economy's capital
stock continue to grow. Ultimately, a technological limit is reached, beyond
which there is no further decline in the cost of renewable energy.

\section{Policy Scenarios}

In this section we consider two alternative policies that could be used
to accelerate the adoption of renewable energy in the economy. The first
policy involves taxing investment in fossil fuel technologies. This policy should
 keep the costs of using fossil fuel high, leading to an
acceleration of the adoption of the competing, renewable energy technology.
The second policy is a direct subsidy to R\&D expenditure in the
renewable energy sector.

\subsection{Scenario 1: Tax on Fossil Fuel Energy}

One way of indirectly subsidizing renewable energy might involve imposing a
tax on fossil fuels. Here, we consider different scenarios regarding the
size of such a tax and explore the implications for renewable technology
adoption and growth.

Introducing taxes on $n$ during the fossil fuel regime, and returning the revenue to households in lump sum form, the budget
constraint in that regime becomes:
\begin{equation}
c+i+n(1+\tau _{n})+g(S,N)R=y+T
\end{equation}
with a corresponding budget constraint for the government given by:
\begin{equation}
\tau _{n}n=T
\end{equation}
The revenue payment from the tax is lump sum in the sense that, when choosing investment in $n$, a private sector decision-maker takes account of the fact that higher $n$ implies a higher tax liability, but $T$ is taken as independent of any one individual's investment decision $n$.The budget constraint in Regimes 2 and 3 is the same as before, so the
analysis of those regimes remains unchanged.

The current value Hamiltonian and thus Lagrangian is now given by 
\begin{equation}
\begin{split}
\mathcal{H}=&\frac{c^{1-\gamma }}{1-\gamma }+\lambda \left[
Ak+T-c-i-j-n(1+\tau _{n})-g(S,N)R-(\Gamma _{1}+H)^{-\alpha }B\right] 
\\ & +\epsilon (R+B-Ak) 
+q(i-\delta k)+\eta B(1+\psi j)+\sigma QR+
\\& \nu n+\mu j+\omega n+\xi R+\zeta
B+\chi \lbrack \text{$\Gamma _{2}{}^{-1/\alpha }-$}\Gamma _{1}-H]
\end{split}
\end{equation}

The first order conditions for a maximum with respect to the control
variables are the same as previously except for $n$, which changes to: 
\begin{equation}
\frac{\partial \mathcal{H}}{\partial n}=-\lambda (1+\tau _{n})+\nu +\omega
=0,\omega n=0,\omega \geq 0,n\geq 0
\end{equation}%

The differential equations for the co-state variables remain as before. The
shadow price of energy will again be 
\begin{equation}
\epsilon =\lambda g(S,N)-\sigma Q
\end{equation}

As before, we also assume parameter values and taxes are chosen so that investment in
fossil fuel technology is productive, that is, $n>0$. Then $\omega =0$ and
hence $\nu =\lambda (1+\tau _{n})$, and, since $q=\lambda $, we have $\nu
=q(1+\tau _{n})$. But then using $\dot{q}=\dot{\lambda}$, and after substituting $R=Ak$, we now obtain 
\begin{equation}
\left[ \frac{\delta }{A}+g(S,N)-\frac{1}{(1+\tau _{n})}\frac{\partial g}{%
\partial N}k-1\right] \lambda =\sigma Q
\label{eq:qdoteqlamdotTaxes}
\end{equation}

Differentiating \eqref{eq:qdoteqlamdotTaxes} with respect to time, substituting for $\dot{N},\dot{\lambda}/\lambda =\dot{\nu}/\nu ,\dot{S},\dot{\sigma}$ and $\dot{Q}=\pi Q$, we obtain a condition
relating the two types of investments ($i$ and $n$) in the initial fossil
fuel economy: 
\begin{equation}
\lambda \left[ \frac{\partial g}{\partial N}\biggl (n(1+\tau_n)+\delta k+\frac{\sigma QAk}{%
\lambda }-i\biggr )-\frac{\partial ^{2}g}{\partial S\partial N}QAk^{2}-\frac{%
\partial ^{2}g}{\partial N^{2}}nk\right] =\sigma\pi Q(1+\tau_n)
\label{eq:i_nTax}
\end{equation}

A second relationship between $i$ and $n$ is given by the budget constraint, which in this regime is:%
\begin{equation}
c+i+n(1+\tau _{n})+g(S,N)R=y+T
\label{eq:BudgetTaxes}
\end{equation}
In equilibrium, however, the government budget constraint will imply that per capita lump sum transfers equal per capita tax revenue. Also, $y=Ak=R$ and $c=\lambda^{-1/\gamma}$, so \eqref{eq:BudgetTaxes} can be written as:
\begin{equation}
i+n=Ak(1-g(S,N))-\lambda^{-1/\gamma}
\label{eq:BudgetTaxEq}
\end{equation}

Substituting \eqref{eq:BudgetTaxEq} into \eqref{eq:i_nTax}, we obtain a modified
equation to be solved for energy technology investment $n$ in the fossil
fuel regime with taxes on such investment: 
\begin{equation}
\begin{split}
& n\lambda\left( \frac{\partial ^{2}g}{\partial N^{2}}k-(2+\tau_n)\frac{\partial g}{\partial N}\right) =\\
&\lambda \left[ \frac{\partial g}{\partial N}[k(\delta
+g(S,N)A-A+\frac{\sigma QA}{\lambda })+\lambda ^{-1/\gamma }]-\frac{\partial^{2}g}{\partial S\partial N}QAk^{2}\right] -(1+\tau_n)\sigma \pi Q  
\end{split}
\label{eq:n_soln}
\end{equation}
Having obtained $n$, \eqref{eq:BudgetTaxEq} determine $i$. The differential equations in the fossil fuel regime remain unchanged.

We consider different scenarios regarding the size of the tax. We summarize
our findings in Table~\ref{table:Tax}. The rows give the date of the transition to the
renewable energy regime ($T_{1}$), the cumulative investment in fossil fuel technology
at that time ($N$), the cumulative exploitation of fossil fuels before they are abandoned ($S$), and the date of transition to the final renewable energy
regime ($T_{2}$). The first column gives the outcome in the absence of government intervention. The next
two columns give the equilibrium values of the same variables when there
is a $2\%$ and a $4\%$ tax on investment in fossil fuel technologies.

\begin{table}[htdp]
\caption{Values of key variables with fossil fuel taxes}
\begin{center}\begin{tabular}{c|cccc}
  & $\tau _{n}=0 $& $\tau _{n}=0.02$ & $\tau _{n}=0.04$\\
\hline
$T_0$ & 42.8775 & -- & -- \\
$k(T_0)$ & 67.0626 & -- & -- \\
$N(T_0)$ & 5.6117 & -- & -- \\
$S(T_0)$ & 219.2506  & -- & --\\
$T_1$ & 44.0729 & -- & -- \\
$T_{2}$ & 57.882 & -- & --\\
$k(T_2)$ & 1940.7248 & -- & --
\end{tabular}
\end{center}
\label{table:Tax}
\end{table}

These findings have a number of implications for policy. First, taxing
fossil fuels accelerates the rate of adoption of the renewable energy
technology. However, it is worth noting that the elasticity of the adoption
rate appears to be small. A tax of $2\%$ reduces $T_0$ by only $1.26\%$. On the other hand, the same $2\%$ tax on $n$ decreases the cumulative extraction of fossil fuel by $7.42\%$, and cumulative investment in fossil fuel technology at the switch date by $8.15\%$. That is, the tax causes fossil fuel reserves to be used
less intensively in the fossil fuel economy in addition to accelerating the transition to renewable energy. This outcome comes at some cost. The distortion created by
the tax creates a wedge between the equilibrium and the socially optimal
level of investment. The capital stock at the time of the transition to renewable energy is $10.38\%$ lower following the imposition of the $2\%$ tax on $n$. More importantly, it can be shown that social welfare in the
economy declines as a result of the tax.\footnote{Absent any government intervention, the \textit{First Welfare Theorem} holds
in our model economy. If, as a result of externalities or other distortions
the First welfare Theorem was to fail, then government policy could become
beneficial.} Perhaps because the capital stock, and thus overall output, is lower under the tax, it takes longer before investment in renewable R\&D becomes positive. It also takes longer before the economy reaches the stationary state where renewable technology attains its maximum feasible level of efficiency.

\subsection{Scenario 2: Subsidy for Renewable Energy}

We are again interested in how effective the subsidy is in both bringing forward the time of the transition
to the renewable energy regime and also in reducing the total consumption of fossil fuels before that time is reached. Introducing a subsidy on $j$ during the regime where $j>0$, the budget constraint in that regime becomes becomes:
\begin{equation}
c+i+j(1-\tau _{j})+pB=y-T
\end{equation}
with a corresponding budget constraint for the government given by:
\begin{equation}
\tau _{j}j=T
\end{equation}
Once again, the tax required to pay the subsidy is lump sum in the sense that individual decision-makers do not believe that their own choices of $j$ will affect their per capita tax bill. The budget constraints in the fossil fuel regime, or when $j=0$, are the same as before, so the analysis of those regimes remains unchanged.

The current value Hamiltonian and thus Lagrangian is now given by
\begin{equation}
\begin{split}
\mathcal{H}& =\frac{c^{1-\gamma }}{1-\gamma }+\lambda \left [
Ak-T-c-i-j(1-\tau _{j})-n-g(S,N)R-(\Gamma _{1}+H)^{-\alpha }B\right] \\
& +\epsilon (R+B-Ak)+q(i-\delta k)+\eta B(1+\psi j)+\sigma QR \\
& +\nu n+\mu j+\omega n+\xi R+\zeta B+\chi \lbrack \text{$\Gamma
_{2}{}^{-1/\alpha }-$}\Gamma _{1}-H]
\end{split}%
\end{equation}

The first order conditions for a maximum with respect to the control
variables once again are the same as before except for $j$ where the condition changes to: 
\begin{equation}
\frac{\partial \mathcal{H}}{\partial j}=-\lambda (1-\tau _{j})+\eta\psi B+\mu =0;\mu j=0,\mu \geq 0,j\geq 0
\end{equation}%
The differential equations for the state and co-state variables remain as before.

Following the previous analysis, in regime 3 with $B=Ak>0$ and $j>0$, we will now have $q=\lambda =\eta\psi Ak/(1-\tau _{j})$, so the shadow price of energy becomes 
\begin{equation}
\epsilon =\lambda(\Gamma_1+H)^{-\alpha }-\frac{\lambda(1+\psi j)(1-\tau _{j})}{\psi Ak}
\end{equation}%
Noting that $q=\lambda $ implies $\dot{q}=\dot{\lambda}$, we obtain 
\begin{equation}
\frac{\dot{\lambda}}{\lambda }=\beta +\delta -A(1-(\Gamma_1+H)^{-\alpha })-\frac{1-\tau _{j}}{\psi k}-\frac{j(1-\tau _{j})}{k}
\label{eq:lamdotSubs1}
\end{equation}
Differentiating $\lambda(1-\tau _{j}) =\eta\psi Ak$ with respect to time, we obtain $(1-\tau_j)\dot{\lambda}=\psi A(\dot{\eta}k+\eta\dot{k})$. Using \eqref{eq:etadot}, $\lambda(1-\tau _{j}) =\eta\psi Ak$ and $\dot{k}=i-\delta k$, we obtain 
\begin{equation}
\frac{\dot{\lambda}}{\lambda }=\beta-\delta -\frac{\alpha\psi (\Gamma_1+H)^{-\alpha
-1}(Ak)^2}{1-\tau _{j}}+\frac{i}{k}
\label{eq:lamdotSubs2}
\end{equation}
Equating \eqref{eq:lamdotSubs1} and \eqref{eq:lamdotSubs2}, we obtain an expression involving the two investments $i$ and $j$ as a function of $k$ and $H$ 
\begin{equation}
i+j(1-\tau_j)=2\delta k-\frac{1-\tau_j}{\psi}-Ak(1-(\Gamma _{1}+H)^{-\alpha})+\frac{\alpha \psi A^2k^{3}(\Gamma _{1}+H)^{-\alpha -1}}{1-\tau_j}
\label{eq:ijeqnSubs}
\end{equation}
The budget constraint and the first order condition for $c$ then provide a
second equation: 
\begin{equation}
i+j=Ak(1-(\Gamma_1+H)^{-\alpha })-\lambda ^{-1/\gamma }
\label{eq:BudgetSubs}
\end{equation}%
where we have once again used the government budget constraint to eliminate the subsidy variable in equilibrium.

For $\tau_j\ne 0$, \eqref{eq:BudgetSubs} and \eqref{eq:ijeqnSubs} can now be solved for $j$ as a function of $H, k$ and $\lambda$: 
\begin{equation}
\tau_jj=2k[A(1-(\Gamma_1+H)^{-\alpha }) -\delta]+\frac{1-\tau_j}{\psi}-\lambda
^{-1/\gamma}-\frac{\alpha\psi A^2k^3(\Gamma_1+H)^{-\alpha -1}}{(1-\tau _{j})}
\label{eq:jSubs}
\end{equation}
with $i$ then given from \eqref{eq:BudgetSubs}. Observe that the higher the subsidy rate $\tau_j$ the more positive has to be the right of \eqref{eq:jSubs}. In turn, this will require a larger value of $H$ for given values of $k$ and $\lambda$. Not surprisingly, we conclude that a subsidy must increase investment in renewable technology knowledge $H$. With $H$ higher, the transition times must also come earlier in time under the subsidy policy. Comparing \eqref{eq:lamdotSubs1} with \eqref{eq:lamdotreg2}, we also see that the renewable R\&D subsidy will alter the differential equation governing the evolution of $\lambda$.

We also should note that, although the equations for the regime where $B=Ak>0$ and $j=0$ are not effected by the subsidy to $j$ the transition to the regime with $j>0$ will be affected. Specifically, the non-negativity constraint on $j$ will now be binding where $\lambda(1-\tau _{j}) =\eta\psi Ak$ rather than $\lambda =\eta\psi Ak$.

As with the tax policy, we consider different scenarios regarding the size of the
subsidy. The first column in the Table~\ref{table:Subsidy} remains unchanged as it
gives the date of the transition to the renewable energy regime ($T_{1}$),
the cumulative investment in fossil fuel extraction at that time ($N$), the cumulative exploitation of fossil fuels at that time ($S$), and the date of
transition to the final renewable energy regime ($T_{2}$) in the absence of
any government intervention. The next two columns give the equilibrium
values of the same variables when there is a $2\%$, and a $4\%$
subsidy on investment associated with renewable energy.

\begin{table}[htdp]
\caption{Values of key variables with renewable investment subsidies}
\begin{center}\begin{tabular}{c|cccc}
  & $\tau _{n}=0 $& $\tau _{n}=0.02$ & $\tau _{n}=0.05$ & $\tau _{n}=0.2$\\
\hline
$T_{1}$ & 51.2249 & 32.4124 & 24.0542 & 15.9956 \\
$N(T_{1})$ & 64.6412 & 87.1836 & 92.4245 & 110.4229 \\
$S(T_1)$ & 382.9009  & 478.2624 & 498.5666 & 566.7097 \\
$T_{2}$ & 131.4168 & 102.4820 & 90.0362 & 75.5973
\end{tabular}
\end{center}
\label{table:Subsidy}
\end{table}

These results contain some useful information for policy. First, a subsidy
on investment in renewable energy accelerates the rate of adoption of the
renewable energy technology. Indeed, although it is hard to compare the two
directly, a renewable energy subsidy appears to be more effective than a tax
on fossil fuels, with a $2\%$ subsidy accelerating $T_{1}$ by $19$ years.
Another important difference with the previous tax scenario is that the
fossil fuel reserves are used more intensively as a result of the subsidy.
The intuition of this result is as follows. Since the adoption of renewable
fuel is accelerated as a result of the subsidy, the opportunity cost of
using fossil fuel in the short run declines. Thus, while the subsidy on
renewables leads to a faster transition away from fossil fuels, it also
implies a more intensive use of fossil fuel than what is socially optimal in
the short run. While we do not model carbon dioxide or other emissions associated with the combustion of fossil fuels explicitly in our
analysis, it is worth mentioning that this could imply an increase in
such emissions in the short run.

\section{Conclusion}

With over two trillion dollars in annual sales, the energy industry is the
largest on the planet. Thus, economic policies that affect the energy sector
have global consequences. Yet, seldom are such policies studied and
evaluated using the standard tools of macroeconomics. In this paper we built
a model in which there is a potential for technological progress in
renewable energy to play the role of an engine of macroeconomic growth. We
computed the equilibrium optimal path of investment in both the fossil fuel and the
renewable energy sectors and calibrated the model to fit current global conditions using data from a variety of sources.
Finally, we evaluated different policy scenarios regarding imposing taxes on
the use of fossil fuel and offering government subsidies on the development
of renewable energy.

We found that, absent any government intervention, the economy goes through
three distinct regimes related to energy production. Initially, production
uses fossil fuel only, and investment takes place in order to improve the
efficiency of supplying fossil fuel. In the medium to long run, as the price
of fossil fuel inevitably increases, investment and capital accumulation
slow down. The economy then makes a transition to the first of two renewable
energy regimes. Subsequently, learning-by-doing reduces the cost of
producing capital using the backstop technology. Finally, in the very long
run, a transition to the second renewable energy scheme occurs. Here, a
limit is reached after which renewable energy is produced at the lowest
possible cost.

We then examined how these transitions are affected by imposing taxes on
fossil fuel and subsidies on R\&D in renewable energy. We found that taxing
fossil fuels accelerates the rate of adoption of the renewable energy
technology. However, a main finding of our analysis is that the elasticity
of the adoption rate appears to be small. In our model economy, a tax as
high as $20\%$ accelerates the renewable technology adoption by about eleven
years, while a more realistic $2\%$ tax accelerates the transition by only
five years. The tax has the additional effect of a less intensive fossil
fuel use. However, the distortion created by the tax creates a wedge between
the equilibrium and the socially optimal level of investment. Hence, welfare
in the model-economy declines in the tax size.

In our model, subsidies on renewable energy investment also accelerate the rate
of adoption of the renewable energy technology. Indeed, a renewable energy
subsidy appears to be more effective than a tax on fossil fuels, with a $2\%$
subsidy accelerating the introduction of the renewable energy regime by
nineteen years. As a result of the renewable energy subsidy, the fossil fuel
reserves are used more intensively in the short run. This somewhat
paradoxical conclusion can be explained as follows. Since the adoption of
renewable fuel is accelerated as a result of the subsidy, the opportunity
cost of using fossil fuel in the short run declines. Thus, while the subsidy
on renewables leads to a faster transition away from fossil fuels, it also
implies a more intensive use of fossil fuel than what is socially optimal in
the short run. While we do not  explicitly model emissions associated with fossil fuel combustion in our
analysis, it is worth mentioning that this could imply an increase in
negative externalities associated with such emissions in the short run.

Our analysis can be extended in many ways. Introducing technology specific
capital could allow us to more accurately capture the trade-off between
fossil versus renewable energy production. Separating out the effects of learning by doing and explicit investment in R\&D
would allow us to capture innovation and cost reduction in the supply of
renewable energy in greater generality. Studying decentralized allocations
will permit us to explicitly account for creative destruction and the
possibility of under-investment in R\&D. Finally, our current calibration
could be modified to target the economy's initial capital stock. This will
allow us to perform more meaningful welfare comparisons across different
policy regimes. We leave these issues to future research. We believe that
our main findings will remain qualitatively true under such extensions,
which we leave to future research.

\newpage

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